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Nanomaterials 2020, 10, 767 11 of 15 Table 3. Relative differences (Equation (14)) of the excess energy per atom (Equation (13)) shown in Figures 5 and 6. D [%] of the Excess Energy Eex Per Atom from the Murnaghan in Figure 5d from the bulk modulus in Figure 5d from the bulk value in Figure 5d spherical shape in Figure 6d decahedral shape in Figure 6d tetrahedral shape in Figure 6d Number of Atoms in the Nanocluster/Nanoparticle 1 7 23 54 105 181 −32.7 −7.5 −6.3 −4.3 −0.7 −2.5 −40.1 −10.9 −8.0 −5.3 −1.4 −3.0 −22.7 0.5 −0.8 0.0 2.7 0.2 −36.6 −17.6 −18.6 −18.0 −15.7 −17.8 −22.7 0.5 −0.8 0.0 2.7 0.2 −10.0 16.9 15.5 16.4 16.6 16.7 Regarding the tetrahedron, the {111} surface energy was used similarly as in the case of the decahedron. As seen in Figure 6a,b, the best agreement between the total energies of nanoparticles determined from phenomenological thermodynamic modeling and those obtained from direct DFT calculations of nanoparticles is found when the actual (decahedral) shape of nanoparticles is considered. Our calculations also confirm the previous findings that the surface energy and strain energy change rapidly with the change of structure (accompanying the change of temperature [8]). In agreement with results published by Vollath et al. in Ref. [62] our analysis also demonstrates that changes of surface energy are not noticeable for nanocluster/nanoparticles sizes less than 4 Å. Finally, it is worth mentioning that our conclusion, that the energy of Ag nanoclusters and nanoparticles can be quite reliably assessed using the volume of the bulk fcc Ag, agrees well with the concept of so-called surface area correction [61,63] which is related to the expansion of the electronic cloud around the nanoclusters/nanoparticles. This phenomenon is specifically important for small nanoparticles and the resulting volume is, in the case of nano-sized systems, put equal to that of the bulk material. Our recommended choice of the volume in the case of nanoclusters/nanoparticles (to be set equal to the volume of the same number of atoms in the bulk) is thus neatly justified also by the electronic structure of the discussed nanoparticles. 4. Conclusions With the help of first-principles calculations, we investigated properties of silver decahedral nanoclusters/nanoparticles containing 1–181 atoms in their static atomic configurations corresponding to the minimum of the quantum-mechanically computed total energies. Our T = 0 K thermodynamic analysis compares excess energies (per atom) obtained from a phenomenological approach, which is mostly based on bulk-related properties, with excess energies of direct quantum-mechanical DFT calculations of actual nanoclusters/nanoparticles. We show that the phenomenological thermodynamic modeling, which uses (i) the bulk reference energy, (ii) surface energies obtained for infinite planar (bulk-related) surfaces and (iii) the bulk atomic volume can predict the excess energy per atom of the studied nanoclusters/nanoparticles with the error lower than 3% with the only exception being the limiting case of a single silver atom. This agreement is achieved when the bulk-related properties (the bulk reference energy, the atomic volume and surface energy) are determined by the ab initio calculations performed as much as possible on equal footing with direct quantum-mechanical calculations of the studied nanoclusters/nanoparticles, i.e., with the same computational parameters (the same exchange-correlation functional, energy cut-off, k-point density, . . . ). The only necessary information related to the nanoclusters/nanoparticles of interest, which enters the phenomenological thermodynamic modeling, is the number of atoms in the nanocluster/nanoparticle, their shape and the crystallographic orientations of facets. Importantly, the quantum-mechanical calculations of bulk-related properties are much less computationally demanding and we demonstrate that a top-downPDF Image | Quantum-Mechanical of the Energetics of Silver Decahedron Nanoparticles
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